Overview

Advanced geometry blends similarity, power of a point, and homothety. Draw accurate diagrams and track equal angles carefully.

Key Ideas

• Ceva's theorem: for concurrent cevians, $\frac{AF}{FB}\cdot\frac{BD}{DC}\cdot\frac{CE}{EA}=1$.
• Homothety maps circles to circles and preserves tangency.
• Power of a point applies to tangents and secants.

Core Skills

Apply Ceva and Menelaus

Use Ceva for concurrency and Menelaus for collinearity. Keep segment order consistent to avoid sign errors.

Use Homothety Centers

Identify the homothety center to map tangency points and parallel lines.

Combine with Similarity

Look for similar triangles created by tangents, secants, or parallel lines.

Worked Example

If two tangents from $P$ touch a circle at $A$ and $B$, then $PA=PB$. This follows from equal tangents to a circle.

More Examples

Example 1: Ceva

If $\frac{AF}{FB}=2$, $\frac{BD}{DC}=3$, find $\frac{CE}{EA}$ for concurrency.

$\frac{CE}{EA}=\frac{1}{6}$.

Example 2: Power of a Point

From $P$, a secant meets the circle at $A,B$ with $PA=3$, $PB=12$. Find the tangent length.

$PT^2=36$, so $PT=6$.

Example 3: Homothety

If two circles are externally tangent at $T$, the line joining centers passes through $T$. This is the homothety center.

Strategy Checklist

• Decide if Ceva/Menelaus applies.
• Look for homothety centers to relate segments.
• Use power of a point to connect lengths.

Practice Problems